2Question: A mathematician studying algebraic topology explores symmetries of a surface modeled by a regular polygon with rotational symmetry of order $ n $. If the polygon has 12 sides, what is the sum of all distinct rotational symmetries (in degrees) that map the polygon onto itself?

Sat, 02 May 2026 20:14:54 +0000

To solve this problem, we begin by understanding that a regular polygon with 12 sides exhibits rotational symmetry. The rotational symmetry group of a regular $ n $-gon includes all rotations by multiples of $ \frac{360^\circ}{n} $ that map the polygon onto itself. Given the polygon has 12 sides, the full rotational symmetry is generated by rotations of: $$
\frac{360^\circ}{12} = 30^\circ
$$ Thus, the distinct rotational symmetries are: